(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Druet, Pierre-Etienne
We consider the problem to optimize the stationary temperature distribution and the
equilibrium shape of the solid-liquid interface in a two-phase system subject to a temperature
gradient. The interface satisfies the minimization principle of the free energy, while
the temperature is solving the heat equation with a radiation boundary conditions at the
outer wall. Under the condition that the temperature gradient is uniformly negative in the
direction of crystallization, the interface is expected to have a global graph representation.
We reformulate this condition as a pointwise constraint on the gradient of the state, and
we derive the first order optimality system for a class of objective functionals that account
for the second surface derivatives, and for the surface temperature gradient.
